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I have implemented techniques similar to the technique best described here: http://freespace.virgin.net/hugo.elias/graphics/x_water.htm. Unfortunately, these quickly result in high frequency noise because of the initial discontinuity of perturbing a single particle. However, though the particles are oscillating, I can make out slow-moving waves. It's these slow-moving waves that I want!

Basically, I'm looking for a hackish algorithm that can generate plausible-looking waves on the surface of a 1D grid, using any detail level of grid.

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I've made a quick implementation with unity here and I didn't like it.. very hard to find the a suitable set or parameters to make the simulation behave properly. It's preytty fast though! My line has 50 points and it runs with more than a thousand fps. I used a damped wave equation and euler method. If you're interested I can give you the code.

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in first place in this method you must limit fps inside your game loop for example if game runs 60-1000 fps water algorithm must run on 40 fps , to fast fps can generate to fast waves = as you say noise.
theres game atempt to use water waveshttp://serumas.gamedev.lt/index.php?id=bakterijos1
page not in english but it contains video of game

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in first place in this method you must limit fps inside your game loop for example if game runs 60-1000 fps water algorithm must run on 40 fps , to fast fps can generate to fast waves = as you say noise.
theres game atempt to use water waveshttp://serumas.gamedev.lt/index.php?id=bakterijos1
page not in english but it contains video of game